# Math Thread



## Hiikaru (Mar 31, 2014)

Look, a math thread!

So, what do you think of math? Are you good at it? What's your favourite (or least favourite) kind of math? What math did you do last? Pi or tau? (hint: pick tau.)

I'm really interested in recreational math! So like, hexaflexagons, drawing infinity elephants, chicken nugget math, that kind of thing! Also everyone should watch those videos. Even if you hate math. _Especially_ if you hate math. Recreational math is why I like math now.

I also really like trigonometry. You basically just draw triangles on everything and then you win. There are some other things like cosines and tangents, but mostly you just draw triangles on everything. Triangle parties! I like trigonometry because you measure an angle on the triangle and do all this interesting puzzle-y stuff to figure out the other angles. And some angles of other triangles that are hanging around. You can do an angle game with just geometry but then you can also add trigonometry for more excitement.

The last kind of math I learned about was happy numbers! You take any integer and add up the squares of its digits. So like 26 you'd do 2 squared + 6 squared. Which is 40. Then you do the same thing to 40! You keep going until you either discover an infinite loop or 1. If you discover 1, you win! It's a happy number. If it's an infinite loop, it's a saaaad number and you have to pick a new number so that you won't be sad anymore. If the number you picked is happy and also prime, it's a happy prime!

I learn about math from Khan Academy, Vi Hart, and Numberphile. Vi Hart is good because you barely even notice there's any math and also she talks all the time about how boring math class is. Khan Academy has a bunch of videos teaching math. Numberphile is about different kinds of numbers and things.

I'm sure everyone has lots of aggressive opinions about math, so, discuss!


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## Karousever (Mar 31, 2014)

I am failing Pre-Calculus.

But I used to really like making everything into a percentage.

Mmm, statistics.


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## Vipera Magnifica (Mar 31, 2014)

Math is pretty useful. Algebra, trigonometry, and calculus have become second nature to me, and I often use them to think about real-world problems.

There's some math that is quite a headache for me to work with though (fourier analysis, partial differential equations). I'm not too familiar with some of the recreational math you mentioned, so I'll have to try some of it out.


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## kyeugh (Mar 31, 2014)

Math is something I'd like to be good at!  My brain is very wired that way, but I have a lot of trouble with spatial things.  Geometry isn't really my bag.  I think people that are good at math are really cool, though!  It's kind of fun just to do sometimes; it's challenging, which is good.


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## Murkrow (Apr 1, 2014)

I came across a neat maths game recently! http://www.sciencevsmagic.net/geo/
You can only make circles and straight line and you try to construct shapes such as regular polygons.


I'm always flip-flopping between whether I like pure or applies maths more. I like pure as I'm studying it because knowing mathematical truths is its own reward. To put it another way, in an applied class I might have an equation to solve. I solve it, and get an answer. So what? Even if the equation was given some sort of context like "a car is moving in this direction at this speed..." it isn't a real car. It doesn't feel very rewarding because there's no point to it. When I solve a pure problem, the problem itself is all the context it needs.
Though on the other hand, when I do get a job, it'll probably be applied. So I can't really tell if I'd enjoy it, since it _would_ be a real car.
Another reason I'm not sure which I like best is the difference in difficulty. Areas of pure maths like algebra I find easy to understand, so I enjoy it because I know what's going on. But when it gets hard enough it becomes really horrible because things like analysis uses so much notation that it gets hard to keep track of everything. If you get slightly behind you become lost so quickly.
Applied maths is generally always easy because I don't often need to prove theorems which is what I usually mess up. I can solve equations just fine and I can prove theorems where I have a good idea of all of the assumptions at work. It's just theorems with lots of assumptions hidden behind the definitions I have trouble with.


Also James Grime (you probably know him from Numberphile!) did a guest lecture at my university the other week.
...too bad I'm studying abroad this year.




Vanilla Mongoose said:


> There's some math that is quite a headache for me to work with though (fourier analysis, partial differential equations). I'm not too familiar with some of the recreational math you mentioned, so I'll have to try some of it out.


I've always been fascinated by Fourier analysis. Don't get me wrong, calculating coefficients or performing transforms can be tedious (so is being asked to graph the sum of the first few terms of a Fourier series by hand, ugh) but I absolutely love the theory behind it.


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## Keldeo (Apr 1, 2014)

Math is neat and fun and useful too!

...Er, random mathy anecdote: My geometry teacher made the class draw elephants in our notes so we could remember theorems because elephants never forget. He also promised us mango egg ice-cream if one of us could recite the first twenty digits of phi (no one could). That was a memorable year. 

The last math I learned was probably how airlines use adjacency matrices in flight planning and dynamic ticket-purchasing websites! It was an interesting, educated, and ok discussion until it devolved into arguing about which carrier was the best. Also, tau > pi; this is an aggressive opinion, hopefully.


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## 1. Luftballon (Apr 1, 2014)

Alligates said:


> Also, tau > pi; this is an aggressive opinion, hopefully.


gosh why is that an _opinion_


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## Murkrow (Apr 1, 2014)

What I don't get about the tau argument (or, at least, Vi Hart's original video) is that it's silly that if you have "a pie" you have 2pi of it, and if you have half of a pie, it's one pi.
But since there's no circular foodstuff whose name sounds like tau, why is the "pie" analogy used at all?
And the letter tau is used because it looks a bit like pi. But it looks like half of the letter pi, when it's supposed to denote twice its value? What?


I don't like tau because a lot of results will look more ugly than they already are. Things like the sum of the reciprocals of squares summing to tau squared over 24; pi squared over 6 is much neater.


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## Music Dragon (Apr 1, 2014)

Murkrow said:


> What I don't get about the tau argument (or, at least, Vi Hart's original video) is that it's silly that if you have "a pie" you have 2pi of it, and if you have half of a pie, it's one pi.
> But since there's no circular foodstuff whose name sounds like tau, why is the "pie" analogy used at all?
> And the letter tau is used because it looks a bit like pi. But it looks like half of the letter pi, when it's supposed to denote twice its value? What?
> 
> ...


Forget about the pie, that's beside the point. The idea is that, logically, you'd want a constant that corresponds to one cycle, not half a cycle. One cycle is one tau radians, half a cycle is half a tau radians, three cycles is three tau radians and so on. It makes a lot more sense.

Consider also that pi is the ratio of a circle's circumference to its _diameter_, while tau is the ratio of the circumference to the _radius_. Certainly we speak of radius more often than diameter.

I can also assure you that many important results in mathematics, physics and engineering will look better using tau instead of pi. Sine and cosine periods, Fourier series, and normal distributions spring to mind.

I use pi out of habit, but I would argue in favor of tau any day.


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## Murkrow (Apr 1, 2014)

I know that was the point of the argument, that was basically me complaining about the "pie" analogy. I don't like people making pie = pi jokes in the first place, so when it adds nothing to the argument, I don't see the point.

I always thought Fourier series looked much better with pi. If you look at 2pi-periodic functions like sine and cosine, but centred at zero, the integrals go between -pi and pi. The zeros of those two functions are exactly pi apart so when you work out the Fourier coefficients or are proving what the eigenvalues can be, there's going to be an n*pi term somewhere in there, which I think looks nicer than n*tau/2.
The normal distribution might work better with tau, but the error function works better with pi!

Some things look nicer one way than the other, but I'd much rather have to write pi some of the time and 2pi most of the time, than tau most of the time and tau/2 some of the time. Division doesn't look as nice, especially when written by hand since it either takes up two lines or just looks messy.


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## Scootaloo (Apr 1, 2014)

Heading to Algebra II my first year in high school (next school year) which I'm pretty hyped for; i like algebra.
Algebra and trigonometry are parts about math that i enjoy. I also like recreational math as well.


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## Superbird (Apr 2, 2014)

Hello math people! I like making neat-looking polar graphs by typing numbers into my TI-84.

So I lost my homework packet that I was supposed to teach myself from during spring break. Can someone please explain parametric derivatives (and especially integrals) for me? I would be very grateful. I know that parametric equations are essentially graphs where for every point (x,y), x = x(t) and y = y(t). Do parametrics always start at t=0 unless otherwise noted? I also know that the slope of a parametric equation is (dy/dt)/(dx/dt), so I just take the derivatives of x(t) and y(t) respectively? Also, how do second derivatives and onwards work here? Does it work the same way for integrals, so I would be reverse-engineering the equation? Would I assume that the given x and y are equal to dX/dt and dY/dt respectively, and then I would divide them and take the integral of that?

Basically, I don't quite understand how parametric calculus works. In my defense, I took Pre-Calculus in a month over the summer, and my teacher really skimmed over this material.


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## Vholvek (Apr 2, 2014)

Murkrow said:


> What I don't get about the tau argument (or, at least, Vi Hart's original video) is that it's silly that if you have "a pie" you have 2pi of it, and if you have half of a pie, it's one pi.
> But since there's no circular foodstuff whose name sounds like tau, why is the "pie" analogy used at all?
> And the letter tau is used because it looks a bit like pi. But it looks like half of the letter pi, when it's supposed to denote twice its value? What?
> 
> ...


I _cannot_ stop playing that thingy. See what happens? Here... Ummm... http://www.sciencevsmagic.net/geo/#...5.3L31.3L37.3L8.3L6.5L2.21L9.25L7.16L31.12L37
Other then that, I _love_ math. It's my favorite subject. I am in 7th grade, taking honors Aglebra I, so you can see that I love it. Of pi and tau, I have no idea what tau is, so definitely pi. On my computer, I have a document saved, with 1,000,000 digits of pi. I just got done with quadratic equations (you know, x=-b±√b^2-4ac/2a[I remembered that!]), so yeah, heh heh, I love math.


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## Butterfree (Apr 4, 2014)

Murkrow said:


> Also James Grime (you probably know him from Numberphile!) did a guest lecture at my university the other week.
> ...too bad I'm studying abroad this year.


D:! I feel your pain. I have the biggest nerdcrush on him.

I love math, but Shadey loves it even more. I tried to get him to register and post in this thread, but he said he wouldn't be able to express the coolness of all the math he's studying in writing.

Personally I'm most partial towards number theory and mathematical puzzles.


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## Murkrow (Apr 4, 2014)

Butterfree said:


> D:! I feel your pain. I have the biggest nerdcrush on him.
> 
> I love math, but Shadey loves it even more. I tried to get him to register and post in this thread, but he said he wouldn't be able to express the coolness of all the math he's studying in writing.
> 
> Personally I'm most partial towards number theory and mathematical puzzles.


I'm pretty sure that the latest numberphile video was filmed in the main lecture hall at my university :(
On the other hand I'm pretty sure I've been in the same room as someone he's met before so I'm two steps away from him sort of!

Get him to post anyway! He doesn't need to express the coolness perfectly.

I usually like number theory but eventually integrals get involved and I'm all "I thought this was discrete maths how to integrals make sense?"



Superbird said:


> So I lost my homework packet that I was supposed to teach myself from during spring break. Can someone please explain parametric derivatives (and especially integrals) for me? I would be very grateful. I know that parametric equations are essentially graphs where for every point (x,y), x = x(t) and y = y(t). Do parametrics always start at t=0 unless otherwise noted? I also know that the slope of a parametric equation is (dy/dt)/(dx/dt), so I just take the derivatives of x(t) and y(t) respectively? Also, how do second derivatives and onwards work here? Does it work the same way for integrals, so I would be reverse-engineering the equation? Would I assume that the given x and y are equal to dX/dt and dY/dt respectively, and then I would divide them and take the integral of that?



For a second I thought you were asking about polar derivatives and I really didn't want to reply because I hate the working out of polar derivatives in terms of the Cartesian ones. But you didn't say that so yay I can reply!

I don't work with parametric graphs much so I might get something wrong but I'll try.
t doesn't have to start at 0. Generally it's given what values t can take. Sometimes it doesn't matter, for example
x = cos(t)
y = sin(t) with t between 0 and 2pi
is a circle, but you can have t be betwen -infinity and +infinity and it'll be the same circle

but sometimes it does matter since the system
x = t
y = t
is the same as the line x = y, but it will only be a section of the line depending on what t is. If t can be negative then it's the normal line y = x, but if you start it at 0 it's only half of that line.

As for the slope, yes you work out the derivatives of x(t) and y(t) there. The derivatives written like dy/dx aren't actually fractions but the notation is well thought out. So when you have (dy/dt)/(dx/dt) you can "cancel" the dt and get just dy/dx. That's why it gives you the slope if you divide the derivatives of x and y with respect to t.
(you aren't actually cancelling, it's just convenient to think of it that way. The real reason you can do it because of the chain rule)

As for the second derivative it doesn't work exactly the same. You have to take the x derivative of dy/dx. It's a bit cumbersome to write it here but the wikipedia article shows it at the bottom.



I'm currently doing some numerical analysis homework. Trying, anyway. When you have questions that ask you to make a computation and gives the final answer to work towards, it's nice that you can check it but if you can't get the answer it's quite frustrating. It makes me spend too much time on that question rather than moving on to the next ones and coming back later.


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## Mustardear (Apr 7, 2014)

Murkrow said:


> I'm currently doing some numerical analysis homework. Trying, anyway. When you have questions that ask you to make a computation and gives the final answer to work towards, it's nice that you can check it but if you can't get the answer it's quite frustrating. It makes me spend too much time on that question rather than moving on to the next ones and coming back later.


This screwed me up so much in one of my exams last year. It's even worse when there's a computation that looks really horrific, so I'm split between

doing the computation
looking for a clever way to simplify the computation
moving on to a different question
On the bright side, it's less demoralizing than staring at a question where I have no idea how to even start.

Oh um I'm in my second year of studying (mostly applied) maths at a university and talking about maths on the internet is another one of my hobbies for which my interest is much greater than my skill.

Something cool I learned recently is that the number of riffle shuffles needed to nearly randomize a deck of 52 cards is about 7. Between 5 and 7 shuffles, the distribution of the order of the deck suddenly changes to be very close to discrete uniform (that is, each order of the deck is roughly equally likely). This sudden change is known as the 'cutoff phenomenon' in Markov Chains if anyone is interested. The result for riffle shuffles was proved in the 1980s by David Aldous and Persi Diaconis and I'm told that the proof is quite elegant. I think it's pretty cool that the solution to a real world problem that is quite easy to understand (how many times do I have to shuffle) involves a surprising result and a nice proof. It's like the convergence of everything I love about maths.
Here's a link if you want to know more: http://www.ams.org/samplings/feature-column/fcarc-shuffle


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## Butterfree (Apr 7, 2014)

GUYS

This is one of the most fun and educational Numberphile videos ever. At least in my opinion! Few things have blown my personal mind like back when I learned and properly started to grasp how stupidly simple computers are and how all this near-omnipotent complexity we're working with every day is built on top of the most inane things in layers upon layers of increasing abstraction. It's _amazing_. And this video demonstrates this fact beautifully... using dominoes.


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## Superbird (Apr 7, 2014)

So I just learned today how to find the length of a curve using calculus/integrals, which is actually just the integral of the distance formula. And that made me realize what the distance formula actually is.

...It blows my mind how important the pythagorean theorem is to a large part of the two-dimensional math and calculus I've learned. How exactly did Pythagoras manage to come up with such a fundamental rule of geometry? 

For that matter, does anyone happen to know of a solid proof of the pythagorean theorem from scratch? I'm curious how exactly one could prove such a thing deductively, rather than simply analyzing an evident pattern, or using the law of sines and/or trig identities (which are themselves derived from the pythagorean theorem, no?).


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## Hiikaru (Apr 8, 2014)

Murkrow said:


> I know that was the point of the argument, that was basically me complaining about the "pie" analogy. I don't like people making pie = pi jokes in the first place, so when it adds nothing to the argument, I don't see the point.


It gives you something solid and tangible to think about when comparing tau and pi! Kind of the whole point of Vi Hart's videos is they aren't supposed to get too numbery, so the pies are a good way to avoid that. It's easy to see why you should just be able to do something shaped like 1/8 to get 1/8 if you use a physical cuttable thing instead of writing down tons of numbers.

I don't care for pie/pi jokes either, but I appreciate having simple visuals.



Murkrow said:


> Some things look nicer one way than the other, but I'd much rather have to write pi some of the time and 2pi most of the time, than tau most of the time and tau/2 some of the time. Division doesn't look as nice, especially when written by hand since it either takes up two lines or just looks messy.


If you run into a tau/2 thing, you can still go back to pi! The point is to make math easy, so if suddenly that's not happening, the point of using tau is gone. For instance, pi is supposed to be easier (at least sometimes?) while building because you actually care about the diameter of a circle for a change.

Pi makes a lot of really _basic_ math frustrating, though! Like circles. It gets so you're just memorizing things because 2pi equalling one circle is _not_ intuitive, and that's a problem.

(also there are a lot of tau/2s you can get rid of by doing the equation differently, like in Euler's identity. But really you can just switch to pi any time.)



Murkrow said:


> And the letter tau is used because it looks a bit like pi. But it looks like half of the letter pi, when it's supposed to denote twice its value? What?


I make sense of it by thinking of it as 2pi vs tau. You don't have to multiply by 2 anymore, so it's like cutting off half of the equation. If you go back to imagining pies, you're removing an _entire pie_ like, linguistically. Out of two of them. So it's cut in half! But yeah it is kind of weird that it looks like half of pi.



Butterfree said:


> GUYS
> 
> This is one of the most fun and educational Numberphile videos ever. At least in my opinion! Few things have blown my personal mind like back when I learned and properly started to grasp how stupidly simple computers are and how all this near-omnipotent complexity we're working with every day is built on top of the most inane things in layers upon layers of increasing abstraction. It's _amazing_. And this video demonstrates this fact beautifully... using dominoes.


This is the _best video_. And also the other video where they make the ten thousand domino circuit!! Two ten thousand domino circuits because they do it again to add up bigger numbers.



Vholvek said:


> Of pi and tau, I have no idea what tau is, so definitely pi. On my computer, I have a document saved, with 1,000,000 digits of pi.


Here!

Tau is 2pi! (6.28 and some other stuff) So basically you get rid of all the silly 2pi things that pop up _everywhere_. Which makes circles easier! And lots of other things by extension.



Superbird said:


> So I just learned today how to find the length of a curve using calculus/integrals, which is actually just the integral of the distance formula. And that made me realize what the distance formula actually is.
> 
> ...It blows my mind how important the pythagorean theorem is to a large part of the two-dimensional math and calculus I've learned. How exactly did Pythagoras manage to come up with such a fundamental rule of geometry?
> 
> For that matter, does anyone happen to know of a solid proof of the pythagorean theorem from scratch? I'm curious how exactly one could prove such a thing deductively, rather than simply analyzing an evident pattern, or using the law of sines and/or trig identities (which are themselves derived from the pythagorean theorem, no?).


Proofs! No need for any trigonometry or ultra-careful analysis over a billion triangles; it turns out it's actually pretty easy to prove. You just kind of doodle stuff around your triangle.

Oh, and an origami proof of the Pythagorean theorem.


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## kyeugh (Apr 8, 2014)

Is that what tau is?  In theory, it's probably a lot more helpful.  Two pi is kind of obnoxious.  I've never used it, but even when I'm introduced to it formally, I'll probably still use 2pi out of habit just like I still use IDs over classes out of habit.


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## hopeandjoy (Apr 8, 2014)

I'm going to fail the AP Calc test and have to take Calc 1 in college.

Sometimes I wish I wasn't majoring in a science.


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## Music Dragon (Apr 9, 2014)

I guess I never really introduced myself in my previous post. I'm currently studying engineering physics, though my master's degree is going to be in mathematics (which, I realized after three years, is in many ways more enjoyable than physics). Right now I'm working on my bachelor's thesis in combinatorial game theory, and I am very, very stuck. (Game theory is great for studying disjunctive sums of games, but the game I'm studying never seems to break down into sums...) Hopefully I'll have something to show for it by the end of this semester, but I'm pretty stressed about it.

I'm aiming for a teaching career, so I do some tutoring as well. Right now I'm teaching the first-year students multivariable calculus (and learning a lot as I go). Tomorrow I'm going to introduce them to the method of Lagrange multipliers. How exciting!


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## Murkrow (Apr 10, 2014)

Mustardear said:


> This screwed me up so much in one of my exams last year. It's even worse when there's a computation that looks really horrific, so I'm split between
> 
> doing the computation
> looking for a clever way to simplify the computation
> ...


On the other hand, I do like proof questions that gives you something to work to. "Prove that X is true"
Though that does scare me away from going into academia. If I'm asked to prove something in a homework I assume the question isn't incorrect and so know that it is true. Even if I can't prove it myself I'm at least sure a proof exists. I can't imagine how hard it is to be an actual mathematician who comes up with new proofs. I can imagine trying to prove something for ages only for it to end up being false in the end.


I have a bit of an interest in Markov processes. They seem to have a lot of applications. I can't study it at university since I don't have the prerequisites (I took a bunch of pure modules instead of stats), but I do occasionally watch some lectures on them that MIT put on youtube.




Butterfree said:


> GUYS
> This is one of the most fun and educational Numberphile videos ever. At least in my opinion! Few things have blown my personal mind like back when I learned and properly started to grasp how stupidly simple computers are and how all this near-omnipotent complexity we're working with every day is built on top of the most inane things in layers upon layers of increasing abstraction. It's _amazing_. And this video demonstrates this fact beautifully... using dominoes.


I'd definitely do something like that if I had lots of dominoes and patience. There's probably lots of other zany ways to to computation as well, though I can't think of any myself.
People seem to love making pointless but impressive machines in Minecraft, doing it with dominoes should be a thing since it's much more difficult, slow, and can't be done by writing a computer program to do it for you.
Also I have some dominoes that can attach to a "track" of sorts. As in, they can fall over as usual but they can't slide anywhere, using something like that could prevent accidental knocking over bits you don't want moving. Though if they wanted Guinness involved using something like that might have been considered cheating.




Music Dragon said:


> I guess I never really introduced myself in my previous post. I'm currently studying engineering physics, though my master's degree is going to be in mathematics (which, I realized after three years, is in many ways more enjoyable than physics). Right now I'm working on my bachelor's thesis in combinatorial game theory, and I am very, very stuck. (Game theory is great for studying disjunctive sums of games, but the game I'm studying never seems to break down into sums...) Hopefully I'll have something to show for it by the end of this semester, but I'm pretty stressed about it.


What about maths do you like over physics? I always sort of regret not doing more physics! 
Though I also get the same feelings relating to computer science so in the end maths is a bit of a middle ground that I'm glad to be studying after all.


I did a course on game theory last semester. It was interesting but it wasn't quite my thing. I've got to wonder though how you come up with payoffs? We learned lots of methods to work out who things like what player has more sway or something like that but a lot of questions we were given provided payoffs for us to use. I'd imagine there are some situations where working out what the payoffs are would need you to measure something not easily quantifiable?
(I'm probably talking nonsense and sounding stupid to you, aren't I?)





I have to chose what subjects I want to learn next year now. I have a vague idea here and there but it's going to take a while for me to decide entirely. The annoying thing about looking through descriptions for courses is that if I don't know the words it uses it makes me think it's too hard for me. But I don't know them because I haven't learned them yet! When I do know the words I assume they're too easy. I don't know what I could do!
It might be easier if I had any idea at all what I want to do when I grow up.
The kind of maths I enjoy constantly changes too. One day I prefer algebra, then it's topology, then it's number theory. So far the only thing I've decided I'm definitely going to do next year is cryptography.



EDIT: Also, another maths thing I want to say. I know the proofs and I understand them, but no matter how hard I try I just can't get my head around the fact that:
1) The number of real numbers is greater than the number of rational numbers
2) The rational numbers are dense in the real numbers
_At the same time_. Both individually make sense to me. Putting them together make my head hurt.


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## Mustardear (Apr 10, 2014)

Murkrow said:


> On the other hand, I do like proof questions that gives you something to work to. "Prove that X is true"
> Though that does scare me away from going into academia. If I'm asked to prove something in a homework I assume the question isn't incorrect and so know that it is true. Even if I can't prove it myself I'm at least sure a proof exists. I can't imagine how hard it is to be an actual mathematician who comes up with new proofs. I can imagine trying to prove something for ages only for it to end up being false in the end.


Reminds me of a quote from Julia Robinson, which perhaps you've heard, when asked to describe what she does during the day:
"Monday:	Try to prove theorem
Tuesday:	Try to prove theorem
Wednesday:Try to prove theorem
Thursday:	Try to prove theorem
Friday:	Theorem false"


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## Music Dragon (Apr 10, 2014)

Murkrow said:


> On the other hand, I do like proof questions that gives you something to work to. "Prove that X is true"
> Though that does scare me away from going into academia. If I'm asked to prove something in a homework I assume the question isn't incorrect and so know that it is true. Even if I can't prove it myself I'm at least sure a proof exists. I can't imagine how hard it is to be an actual mathematician who comes up with new proofs. I can imagine trying to prove something for ages only for it to end up being false in the end.


Don't let that put you off! Most people seem to believe that being a mathematician means being really really smart until poof, theorems. But the truth is that it doesn't take a sudden flash of inspired genius to produce results. You can actually tackle the unknown in a somewhat systematic fashion. After analyzing a problem for a while, you might come up with a super-interesting conjecture that you're unable to prove, or you might be able to prove something that seems rather trivial and insignificant, or - as you said - you might spend ages trying to prove something that turns out to be false in the end. But all of that is still progress! This is true of the advancement of _any_ field of study, I think. You don't have to just wake up one day and invent Galois theory. These things happen in small steps, and with lots of collaboration. Anything you discover, anything at all, is a step in the right direction.



Murkrow said:


> What about maths do you like over physics? I always sort of regret not doing more physics!
> Though I also get the same feelings relating to computer science so in the end maths is a bit of a middle ground that I'm glad to be studying after all.


I don't rightly know! I guess I just started noticing after a while that I was skimming over all the physics courses because I was so engrossed in mathematics. They're both fascinating areas of study, really. Physics is a description of the natural world that we live in, and that's certainly worthy, but I think mathematics goes beyond that. We use mathematics as a tool to solve and understand physical problems, but you can also extend it to things that don't have to exist in reality. It is an art of abstraction and generalization.



Murkrow said:


> I did a course on game theory last semester. It was interesting but it wasn't quite my thing. I've got to wonder though how you come up with payoffs? We learned lots of methods to work out who things like what player has more sway or something like that but a lot of questions we were given provided payoffs for us to use. I'd imagine there are some situations where working out what the payoffs are would need you to measure something not easily quantifiable?
> (I'm probably talking nonsense and sounding stupid to you, aren't I?)


No, not at all. But I think you're talking about classical game theory, as opposed to combinatorial game theory! The classical theory is too close to economics for me, really. I'm always put off by the thought of _applying_ mathematics to practical things...


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